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Harmonic Functions and Harmonic Measure David Mcdonald [email protected]

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Harmonic Functions and Harmonic Measure David Mcdonald Descrimen@Gmail.Com University of Connecticut OpenCommons@UConn Honors Scholar Theses Honors Scholar Program Spring 5-1-2018 Harmonic Functions and Harmonic Measure David McDonald [email protected] Follow this and additional works at: https://opencommons.uconn.edu/srhonors_theses Part of the Partial Differential Equations Commons Recommended Citation McDonald, David, "Harmonic Functions and Harmonic Measure" (2018). Honors Scholar Theses. 558. https://opencommons.uconn.edu/srhonors_theses/558 Harmonic Functions and Harmonic Measure David McDonald B.S., Applied Mathematics An Undergraduate Honors Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of Bachelor of Science at the University of Connecticut May 2018 Copyright by David McDonald May 2018 i APPROVAL PAGE Bachelor of Science Honors Thesis Harmonic Functions and Harmonic Measure Presented by David McDonald, B.S. Applied Mathematics Honors Major Advisor HONORS ADVISOR NAME Honors Thesis Advisor THESIS ADVISOR NAME University of Connecticut May 2018 ii ACKNOWLEDGMENTS I would like to thank Professor Murat Akman for guiding me through a subject that, prior to this year, I was unfamiliar with but interested in. I also would like to thank Professor Keith Conrad for sharing the tex files and guiding me for the process of writing this honors thesis. I'd also like to thank the University of Connecticut for giving me the opportunity to write this thesis, as well as my high school calculus teacher Mrs. Zackeo for teaching me how to do math correctly. Finally I'd like to thank my friends and family for always being there to support me; especially Leo for making sure his thesis is less interesting than mine, Will for raving to SHM with me, Chris Pratt for encouraging me to follow my dreams, Alicia for her assistance in the researching of Ja-Fire, Jacob for always having so much more work than me, and Kelly for helping me continue my streak of never losing at trivia. iii Harmonic Functions and Harmonic Measure David McDonald, B.S. University of Connecticut, May 2018 ABSTRACT The purpose of this thesis is to give a brief introduction to the field of harmonic measure. In order to do this we first introduce a few important properties of harmonic functions and show how to find a Green's function for a given domain. Following this we calculate the harmonic measure for some easy cases and end by examining the connection between harmonic measure and Brownian motion. iv Contents Introduction................................... 1 Ch. 1. Harmonic Functions 2 1.1 Mean Value Property........................... 4 1.1.1 Mean Value Property for Volume................ 5 1.2 The Maximum Principle......................... 6 1.3 The Poisson Integral and Green's function............... 8 1.3.1 Green's Function for the Upper Half Plane........... 15 1.3.2 Green's Function for the Unit Ball ............... 16 1.3.3 Green's Function for the First Quadrant............ 18 1.3.4 Poisson's Formula for the Ball.................. 19 Ch. 2. Harmonic Measure 26 2.1 Examples ................................. 28 2.1.1 The Upper Half Plane ...................... 28 2.1.2 The Unit Disk........................... 30 2.2 Brownian Motion............................. 32 2.2.1 Harmonic Measure in Brownian Motion............. 33 2.2.2 Multiply Connected Domains .................. 35 v Introduction Harmonic functions are those which satisfy Laplace's equation. They have a num- ber of convenient properties including mean value properties and the Maximum Prin- ciple that make them easy to work with. Understanding harmonic functions is the first step to calculating the harmonic measure. The next step is finding Green's func- tions; in section 1.3 we'll introduce the Dirichlet problem and show how to solve a Green's function. To make it less abstract there are a few examples: the upper half plane, the unit ball, and the first quadrant. Then, using the Green's function for the unit ball, Poisson's formula for the Ball can be found. These pieces are all integral to understanding and being able to find the harmonic measure. In probability theory, the harmonic measure ! of a particle in some domain Ω is the probability that it will exit Ω within some E along @Ω. At the beginning of chapter two we'll calculate the harmonic measure for both the ball of radius r and the upper half plane as well as prove its uniqueness and existence in all cases. Harmonic measure has had quite a few recent theorems published, including Makarov's theorem which was quite a breakthrough at the time, so we'll conclude by going over those. 1 Chapter 1 Harmonic Functions Before diving into harmonic measure, it is important to first have some knowledge of what a harmonic function is and the properties that it has. A harmonic function is any function that is twice continuously differentiable and satisfies the Laplace partial differential equation, that is, in Rn, n ≥ 2, n X @2u @2u ∆u = u = + ::: + = 0: xixi @x2 @x2 i=1 1 n Here ∆ is known as the Laplacian and denotes the sum of the second partial deriva- tives for each respective variable. One fundamental property of harmonic functions is that they are invariant under both rotations and translations. That is to say, both rotations and translations of harmonic functions are also harmonic. 2−n 2 2 2 (2−n)=2 Example 1.0.1. We shall see that v(x) = jxj = (x1+x2 :::+xn) is harmonic n 2 2 1=2 2 in R n f0g when n > 2 and v(x) = ln jxj = ln(x1 + x2) is harmonic in R n f0g. 2 Let us start with n > 2, in this case, for i = 1; 2; : : : ; n, we have 2 2 (−n)=2 vxi = (2 − n)xi(x1 + ::: + xn) and 2 2 (−n)=2 2 2 2 (−n−2)=2 vxixi = (2 − n)[(x1 + ::: + xn) − nxi (x1 + ::: + xn) ]: Now n n X X 2 2 (−n)=2 2 2 2 (−n−2)=2 vxixi = (2 − n) [(x1 + ::: + xn) − nxi (x1 + ::: + xn) ] i=1 i=1 n −n (−n−2)=2 X 2 = (2 − n)[njxj − njxj xi ] i=1 = (2 − n)n(jxj−n − jxj−n) = 0: Hence v is harmonic in Rn. 2 2 1=2 When n = 2, i.e when v(x) = ln(x1 + x2) we have 2 xi 1 2xi vxi = 2 2 and vxixi = 2 2 − 2 2 2 : x1 + x2 x1 + x2 (x1 + x2) From this we get 2 2 X X 1 2xi v = [ − ] xixi x2 + x2 (x2 + x2)2 i=1 i=1 1 2 1 2 2 2 2 X = − u2 x2 + x2 (x2 + x2)2 xi 1 2 1 2 i=1 2 2 = 2 2 − 2 2 = 0: x1 + x2 x1 + x2 3 1.1 Mean Value Property Another important property of harmonic functions is called the mean value property. The mean value property states that, given u harmonic on the closed ball centered at a with radius r, B¯(a; r), u is equal to the average of u over @B(a; r). To be more precise, we have Z u(a) = u(a + rζ)dσ(ζ) S where S is the unit sphere and σ is the Borel probability measure on the unit sphere so that σ(S) = 1. Proof. First assume that n > 2 and take B(a; r) = B and 2 (0; 1). Then using Green's identity, it follows that Z Z (u∆v − v∆u)dV = (uDnv − vDnu)ds Ω @Ω with fΩ = x 2 Rn : < jxj < 1g and v(x) = jxj2−n. This results in Z Z Z Z 1−n 2−n 0 = (2 − n) uds − (2 − n) uds − Dnuds − Dnuds (1.1.1) S S S S Here Dn is the differentiation with respect to the outward unit normal, Dn(ξ) = hru(ξ); n(ξ)i. Now since u is harmonic and v ≡ 1 we can apply another form of Green's identity, Z Dnuds = 0: @Ω 4 By the identity above, the last two terms of (1.1.1) are 0, so we are left with Z Z uds = 1−n uds S S which is just Z Z udσ = u(ζ)dσ(ζ): S S This is what we are looking to prove with ! 0 with u continuous at 0. For n = 2 the proof is exactly the same, with the exception that jxj2−n be replaced with log jxj. 1.1.1 Mean Value Property for Volume A similar property exists with respect to the volume of the ball rather than the surface, and its proof utilizes the polar coordinates formula for integration on Rn, 1 Z Z 1 Z fdV = rn−1 f(rζ)dσ(ζ)dr (1.1.2) nV (B) Rn 0 S The mean value property for volume states that if u is harmonic on B¯(a; r); then u(a) is equal to the average of u over B(a; r), or 1 Z u(a) = udV : V (B(a; r)) B(a;r) 5 Proof. Once again take B(a; r) = B and apply equation (1.1.2) setting f to be u multiplied by the characteristic function ξ of B. Following this, simply apply the standard mean value Property to get the desired result. From the mean value property we gain an important insight about the singularities of harmonic functions. Corollary 1.1.1. Any zeros of a real-valued harmonic function are not isolated. Proof. Take u to be real-valued and harmonic on the open connected set Ω with a 2 Ω and u(a) = 0. Now consider the closed ball B¯(a; r) ⊂ Ω with r > 0; since the average of the harmonic function u over the boundary of the ball is 0, u must either be identically 0 or it must take both positive and negative values on @B(a; r).
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